Generalized Normality And The Structure Of Its Critical Groups

In this paper,the properties of a subgroup are studied,that is,the effect of semi 2-subnormal cover avoidance properties on characterizing the solvability of finite groups,and complete classification of the structure of two critical groups is given.First,a semi 2-subnormal cover avoidance subgroup is defined.Let the quotient group/be the subnormal factor of group,is a subgroup of.Ifsatisfies=,thencovers/,if?=?,thenavoidance/.Ifcovers or avoidance every even order factor of a composite column of,thenis called a semi 2-subnormal cover avoidance subgroup of.The minimal counterexamples are used to characterize the solvability ofwith the help of maximal subgroup ofhas semi 2-subnormal cover avoidance properties and solvability of odd order group.Next,the structure of a critical group with certain properties is determined.A set={(2₁,...,(2}of nilpotent Hall subgroups ofis a complete Wielandt set if(|(2₄)|,|(2₅)|)=1 for all4),5).A finite groupis called a GPST-group ifhas a complete Wielandt setsuch that every member inpermutes all maximal subgroups of any non-cyclic subgroupin.In this paper,a complete classification of those groups which are not GPST-groups but all of whose proper subgroups are GPST-groups is given,i.e.,they are precisely minimal non-GPST-groups.Finally,the structure of another critical group is described.A subgroupof a groupis called Hall normally embedded inifis a Hall subgroup of the normal closure.Call a groupan HNE^*-group if every subgroup ofis Hall normally embedded in,and give a complete classification of those groups which are not HNE^*-groups but whose proper subgroups are all HNE^*-groups.